Haloalkyl aryl diphosphates



. 3 phosphorus-containing radical selected from the class consisting ofing of hydrogen and alkyl groups.

from about l C. to about 125 C. but is preferably conducted at atemperature of from about 50 C. to

"about 90 C. A particularly preferred temperature is about 70 C., atwhich temperature the reaction proceeds 'smoothly and with the evolutionof heat. H

Although the reaction'can be conducted under. pressures of from about 1p.s.i.a. to about 150 p. s.i.a., atmospheric pressure is preferred.

Suitable catalysts for the reaction are the'halides of zirconium,aluminum-and'titanium. The preferred catfalyst is titaniumtetrachloride, because it is'hig'hly active ==and easily handled.

The amo'u'nt'of catalyst employed can vary considerselected from theclass consisting of p,p'-alkylidenedi' i The reaction can be conductedat temperatures of was prepared as follows:

'ably. As little as"0.2 percent by weight of the catalyst, based uponthe phosphorus compound, is sufficient for the reaction to proceed. Acatalyst concentration of from 0.7 to 2' percent is preferred for largescale opera- "tions," but larger amounts can be used. Either thetechn'ical grade or the commercially pure grade of the cat- '-alyticcompound can be employed.

The reaction proceeds readily and the addition may be "completedin about1 hour. It is preferred to heat the reactants for a period of at least 5hours. However, the reactants may be heated for longer or shorterperiods of time, depending upon such factors as the actual reactantsused, the size of the charge and the cooling media employed.

om-on-om-o-z-o-om-on-om The following examples are illustrative:

Example 1 0 i} Go omcimoloimo cm) To a solution of 1.0 g. of titaniumtetrachloride and ,120 g. (0.525 mol) of dibutyl chlorophosphate wasadded, dropwise, 85 g. (0.25 mol) of the diglycidyl ether ofp,p'-isopropylidenediphenol. This addition was made at such a ratethat-it was possible to maintain the reaction temperature at 70 C.without either heating or cooling the reaction mixture. After theaddition the reaction mixture was maintained at 70 C. for 6 additional'hours, neutralized using 300 g. of 10 percent aqueous trisodiumcitrate, .washed twice with 300 cc. portions'of water, and stripped bydistillation at a kettle temperature of 120 C. at 3 mm. Hg. There wasobtained 151 g. of clear, pale yellow, viscous liquid as the residueproduct. It was found to have the following properties: acidity=0.l1 cc.of N- base/g., salt=0.005 cc. of N l-IClO /g., n =1.5039, percentCl=8.57 (theory: 8.89), molecular weight=770 (theory=797.7), andyield=75 (based on the diepoxy compound), percent P- 7.60 (theory=7.77).

A tetrabutyl p,p'-isopropylidenediphenoxy-di-(bromopropyl) diphosphatecan also be prepared in this manner.

Example 2 0 ll 0 CHaCzHrClO P (O CaHtCl):

I After completion of the reaction the crude reaction was prepared bythe dropwise addition of 96 g. of the products can be purified bywashing with about an equal volume or more of an aqueous solution,containing from about 5 to 10 percent by weight of trisodium citrate,sodium potassium tartrate or disodium tartrate in order to convert thecatalyst to a water soluble complex and to hydrolyze any excess of themonohalophosphate reactant. An aqueous slurry containing 2 to 10 percentby weight of calcium or barium carbonate may be used instead of theaqueous solution. Following this treatment,

lequation, wherein X is a halogen atom and is a member 15 diglycidylether of 2,2-bis(4-hydroxyphenyl)butane (92.1% material by epoxyanalysis) to an agitated solu tion of 1.0 g. of titanium tetrachlorideand 0.525 mol of di-(Z-chloroethyl) chlorophosphate. It was necessary tocool the reaction mixture throughout the addition in order to maintainthe temperature at 70 C. After the addition the reaction mixture washeated at 70 C. for 6 additional hours, diluted with 200 cc. of benzene,neutralized using 1000 cc. of 10% aqueous trisodium citrate, washed withwater, and stripped by distillation at 70 C./ 1 mm. Hg. There wasobtained 192 g. of yellow, very viscous residue having the followingproperties: acidity=0.006 'cc.

of N base/g., salt=0.03 cc. of N HClO /g., percent Cl=23.76 (theory=24.54), percent yield=84 (based on the diepoxy compound), percentP=7.29 (theory =7.15.).

' The theoretical chlorine and phosphorus contents are based on theassumption that the impurity present in the epoxide starting materialwas present in the final product. Thus, in this owe, the impurity in thefinal product was assumed to be approximately 76 grams -92.1 ;7.0% whichis equivalent to 0.079x96 g.=7.6 g.) i f 2,886,295 "A teas-taamioatu)p,5' biltyuassdlphaiiaiyai t'broinopropyl) diphosphate can also beprepared in this manner. j r

was prepared by the dropwise additionof'dl g. ofthe diglycidyl ether ofresorcinol (90.8% by epoxy analysis) to an agitated solution of 1.0 g.of titanium tetrachloride J V Example 3 and 0.525 mol ofdi-(Z-chloroethyl) chloroph'osphate. It A tetra;- (Z-hloroethyl) thy1iddi h di. 5 was necessary to cool the reaction mixture throughout'iehloropropyl) diphosphate represented by the structural this additionin order to maintain the temperatureat rema n; H 70 C. After theaddition the reaction mixture was f I 43H! W (ClCaHtOhPOCzHsClCHzOOJ-O011202111010P(OCzH4 C )2 H t was prepared by thedropwise addition of 89 g. of-the heated at 70 C. for 6 additionalhours, neutralized using diglycidyl ether of alpha,alpha-bis(4-hydroxyphenyl)- 1000 g. of 10% aqueous tri'sodium citrate,washed with ethane (92.2% by epoxy analysis) to an agitated solution 200cc. of water, and strippedby distillation at 60 C./ 1 of 1.0 g. oftitanium tetrachloride and 0.525 mol of dimm. Hg. The resulting 164 g.of yellow, viscous residue ('2-chloroetliyl) chloroplros'phate. -It was'necessary. -to had the following properties: acidity=0.01 cc. of Nbas'e/ co'olthe reaction mixture throughout'this addition'in order g.,salt=0.008 cc. of N HClO /g., percent Cl'=28.99 tomaintainthetemperature at 70 C.' After the addition (theory =29.2;6), percentyield=90 (based on the diepoxy the reaction mixture was maintained at 70C. for 6 compound), percent P=8.81 (theory =8.52)'. additional hours,diluted with 200 cc. of benzene, new tralized using 1000 cc. of 10%aqueous trisodium citrate, washed with water, and stripped bydistillation at 70 E l 6 C./ 1 mm. Hg. There was obtained 188 g'. ofyellow, xamp e very viscous residue which'had the following pro'pertieszA tetra (2 chloroethyl) p,p'-sultonyldiphenoxy-diacidity=0.01 cc. of Nbase/g., 'salt=0.007' cc. of N (chloropropyl) diphosphate representedbythe structural HClO /g., percent Cl=25.49 (theory =25.42),,perc entformula: v

O was prepared by the dropwise addition of 100 g. of'the propyl)diphosphate can also be prepared in this manner;

yield-#89 (based on the diepoxy compoundL percent P=.7.55'(theory=7.4O). diglycidyl ether of 4,4'-dihydroxydiphenylsulfone (91% ;,:A;-,tetra-(Z-chloroethyl). p,p'-ethyli denediphenoxy-diby epoxy analysis)dissolved in, 400 cc. of ethylene di- (bromopropyl) diphosphate can alsobe prepared in this chloride to an agitated 'solutionof 1.0 g. oftitanium manner. a tetrachloride and 0.525 mol of .di-(Z-chloroethyl)chlorot Example 4 phosphate at a: reaction temperature of 70 C; Afterthe 1 h 1 n 1 dio i.(' hl maddition the reaction mixture was heated at70 Cufor propyl) diphosphate represented by the structural 6 additionalhours, neutralized using 1000 g. of 10% f l A H I 40 aqueous-trisodiumcitrate, washed with 400 .cc. of water,

r o i H II (QICzHiOhP O CzHa (310E200!) C'HzCrHzClO P (O C2H4Cl)a wasprepared by the vdropwise addition of a solution of A5 and stripped bydistillation at 60 C./ 1 mm. Hg. The 57 g. of the diglycidyl ether ofhydroquinone (97.1% by resulting 193 g. of yellow, extremely viscousresidue had epoxy analysis) and 100 cc. of ethylene dichloride to an thefollowing properties: acidity=0.02 cc. of N base/lg, agitated solutionof 1.0 g. of titanium tetrachlorideand salt=0.002 cc. of N. HClO /g.,percent Cl=2 5'.19 0.525 mol of ,di-(2 chloroethyl) chlorophosphate, Itwas (theory 24.11), percent yield=87 (based on the diep'oxy necessary tocool the reaction mixture, in order to maincompound), percent P=7.15(theory ='7.02). I tain the temperature at 70 C. After'the addition. theA tetra- (Z-chloroethyl) p,p-sulfonyldiphenoxy-direaction mixture washeated at 70 C. for, 6 additional (bromopropyl) diphosphate can also beprepared in this hours, neutralized using 1000 g. of 10% aqueoustrisodium manner. citrate, washed with water, and stripped bydistillation at Example 7 4, 70 C./ 1 mm. Hg. There was obtained 159 g.of pale yellow, viscous residue product which had the following A IO YDp,P'- opropyhdenediphenoxyproperties: acidity=0.03 cc. of N base/g.,salt=0.003 cc. di-(chloropropyl) diphosphate represented by the stru erof N HCl0 /g., percent Cl=29.45 (theory =29.92), pertural formula:

II t I ll (OlC:Ht0)rPO CzHzOlCHrOO-(|OO cnlctrnoro 1x00211101) CH3cent-yie1d=89 (based on the diepoxy compound), perwas prepared by thedropwise addition of 85 g. (0.25 centfP=8.93 (theory =8.72). J mol) ofthe diglycidyl ether of p,p '-isopropylidenedi A tetra-(Z-chloroethyl)p-phenyle'nedioxy-di-(bromo phenol'to a flask containing 1.0 g. ofaluminum chloride propyl) diphosphate can also be prepared in thismanner. and 127 g. (0525 mol) of di-(2-chloroethyl) chlorophos- 7 phateover a period of 30 minutes. There was a heat of reaction observedthroughout most of the addition and for.25 minutes afterward whichnecessitated cooling. :of 6 the reaction mixture. After the additionthereaction mixture was heated at C. for 5 additional hours, dilutedwith cc. of ethylene dichloride, neutralized using 300 g. of 10% aqueoustrisodium citrate, washed with 200 cc. of water, and stripped bydistillation; at i 73 100 -C./ 1mm-. Hg. The 179 not viscousliquidExample 5 v r ,tetra -(2-chloroethyl); rn-phenylenedioxy-di(chloroprppyl) diphosphate represented by the structural formula: 1cc'ictmolli oolnrclc'rno- 0enrclnro'1oiuogmcnr A tetra-(Z-chloroethyl)m-phenylenedioxy-di-(bromo A tetra-(Z-chloroethyl)p,p'-isopropylidenediphenoxydi-(chloropropyl) diphosphate represented bythe structural formula:

a After the addition (the 'reaction mixture -was kept at 70' C. for :anadditional 1 hour. The reaction mixture was diluted with 100 cc. ofbenzene, neutralized using 300;. of 10% aqueous trisodium citrate,washed with water, and stripped by distillation at 100 C./ 2 mm. Hg. Theresulting '99 g. of residue was dissolved in 200 cc. of ethyl ether andwashed with dilute aqueous sodium hydroxide, then with sodiumbicarbonate solution, and final- 1y washed "with until neutral towardlitmus paper. The washed ethereal solution was stripped by distillationat 100 C./ 2 mm. Hg. There was obtained 76 g. of

was prepared by the dropwise addition of 85 g. (0.25 mol) of thediglycidyl .ether of p,p'-isopropylidenediphenol to a reaction flaskcontaining 1.0 g. of zirconium tetrachloride and 127 g. (0.525 mol) ofdi-(Z-chloroethyl) chlorophosphate over a period of 15 minutes. It wasnecessary to cool the reaction mixture in order to maintain thetemperature at 70 C. After the addition the reaction mixture was heatedat 70 C. for 1.75 hours, neutralized using 300 g. of 10% aqueoustrisodium citrate, washed with water, and stripped by distillation at100 C./ 1 mm. Hg. The 168 g. of viscous liquid product had the followingproperties: acidity=0.08 cc. of N base/g., salt=0.0l cc. of N HClO /g.,percent Cl=23.38 f( theory='25.83), percent yield=82 (based onthediepoxy compound), percent P=6.64 (theory=7.53).

A tetra-(Z-chloroethyl) p,p'-isopropylidenediphenoxydi-(bromopropyl)diphosphate can also be prepared in this manner.

Example 9 .A =tetra-(2-chl'oroethyl) p,p' isopropylidenediphenoxy-.di-(chloropropyl) diphosphate represented by the struc tural formula:

3 rclcnatoni ocsmolcmoOowas prepared by the dropwise addition of 85 g.(0.25 mol) of the diglycidyl ether of p,p'-isopropylidenediphenol toanagitated solution of 1.0 g. of titanium tetrachloride and 127 g. (0.525mol) of di-(Z-chloroethyl) chlorophosphate. It was necessary to cool thereaction mixture throughout the addition in order to maintain thetemperature at 70 C. After the addition the reaction mixture was heatedat 70 C. for 8 additional hours, neutralized using 300 g. of aqueoustrisodiurn citrate, diluted with 250 cc. of ethyl ether, washed with two100 cc. portions of water, and stripped by distillation at 100 C./ 1 mm.Hg. There was obtained 185 g. of yellow, viscous residue which had thefollowing properties: acidity=0.10 cc. .of N base/g, salt=0.0004 cc. ofN 'HClO /g., percent C1=-24.44 (theory=25.83) and percent yield=90(based on the diepoxy compound).

Example 10 A tetraphenyl p,p'-isopropylidenediphenoxy-di-(chloropropyl)diphosphate represented by the structural formula:

0 (neurone o memento-O o :diphenyl chlorophosphate and 1.0 g. oftitanium tetrachloride over a period of 15 minutes.

It was necessary to cool the reaction mixture throughout the addition inorder tomaintain the reaction temperature at 70 C.

very viscous, yellow liquid residue which had the followme P operties:percent Cl=8.07 (theory=8.08), percent P= 6.7-4.(theory=7.06p), percentC=6 1.37 (theory=6l.52), percent ;H==5.'52 (-theory=5.06), percentyie1d=69, salt=nil, u 1.5729.

Example 11 A'2,2 [p,p isopropy'lidenediphenoxy di (chloropropoxy)Jbis.(5.,5 diethyl 2 oxo 1,3,2 dioxaphosphorinane) represented by thestructural formula:

was prepared by lthfi dropwise addition of 170 g. (0.5 mol) of thediglycidyl ether of p,p'-isopropylidenediphenol to in agitated solutionof 223 g. (1.05 rnols) of 2 chloro -"5,5 diethyl 2 oxo 1,3,2dioxaphosphorinane "and 4.0 g. of titanium tetrachloride over a periodof 45 minutes. 'It was necessary to cool the reaction =mixturethroughout most of this addition in order to maintain the kettletemperature at 70 C. The reaction mixture became so viscous during thisaddition that .100 cc. of ethylene dichloride was added. After theaddition was complete the reaction mixture was heated .at 70 C. for 8additional hours, neutralized by stirring at 25 C. for 1 hour with asolution of 26.6 g. of sodium hydroxide dissolved in cc. of water,washed twice with .250 cc. portions of water, and stripped bydistillation at C./2 mm. Hg. There was obtained 270 g. of yellow, glassy.residue product which had the following properties: acidity=0.004 cc.of N base/g., percent Cl=9.97 (theory=9.27), salt=0.01 cc. of N HClO/g., percent P=7.85 (theory=8.09), percent C=54.25 (theory=154.9),percent H=6.57 (theory-6.84), molecular weight=720 (theory=7 65.6),percent yield=70.

A 2,2 [p,p' isopropylidenediphenoxy di (bromopropoxy)] bis(5,5 diethyl 2oxo 1,3,2 dioxaphos- 'phorinane) can also be prepared in this manner.

Example 12 A 2,2 [p.p' isopropylidenediphenoxy 'di (chloropropoxy)]bis(5 ethyl .2 oxo 4 propyl 1,3,2 dioxaphosphorinane) represented by thestructural formula:

i -o onnowmoQ e was prepared by the dropwise addition of 17.0 g. (0.5mol) of the diglycidyl ether of p,p'-isopropylidenediphenol to anagitated solutionof 237 g. (1.05 mols) of 2 .19 2 chloro 5 ethyl 2 oxo 4propyl 1,3,2 dioxaphosphorinane and *4.0sg.of titanium tetrachlorideo-'ver a period of 45 minutes. It: was necessary to cool the reactionmixture throughout the addition in order to maintain the kettletemperature at 7 0 C. After thea'ddi tion the reaction mixture washeated at' 70 C. for 6' obtained 373 g. of light orange, tacky residuewhich had" the following properties: acidity=0.17 cc. of N base/g,salt=0.03 cc. of N HClO /g., percent Cl=9.43 (theory=8.94), percentC=54.21 (theory=56.0), percent H=7.0 (theory=7.l2), percent P=7.99(theory=7.80), molecular weight=740 (theory=793.7), percent yield=94.

A 2,2 [p,p'-isopropylidenediphenoxy di (bromo propoxy)] bis(5 ethyl 2oxo 4 propyl 1,3,2 dioxaphosphorinane) can also be prepared in thismanner.

Example 13 v A 2,2 [p,p isopropylidenediphenoxy di (chloropropoxy)l bis(4 methyl 2 oxo 1,3,2 7 dioxaphosphorinane) represented by thestructural formula:

om-o 0 cm on. \ioonnoromoG p was prepared by the dropwise addition of170 g. (0.5 mol) of the diglycidyl ether of p,p'-isopropylidenediphenoltoan agitated solution of 179 g. (1.05 mols) of 2 chloro 4 methyl 2 oxo1,3,2 dioxaphosphorinane and 4.0 g. of titanium tetrachloride over aperiod of 45 minutes. It was necessary to cool the reaction mixturethroughout the addition in order to maint'ain the kettle temperature at770 C. Afterthe addition 'the reaction mixture was stirred at 25 C. for2 hours with 600 g. of 15% aqueous sodium citrate, washed with wateruntil neutral toward litmus paper, and stripped by distillation at 100C./2'mm. Hg. There was obtained 310 g. of reddish-brown, very viscousresidue which had an acidity=0.25 cc. of N base/g. This 310 g. ofproduct was dissolved in ethylene dichloride, neu-1 propoxy) 1 'bis(4methyl 2 oxo 1,3,2 dioxaphosphorinane) can also be prepared in thismanner.

-K'Ihe' compounds of this invention are' neutral esters which are-usefulas hydraulic fiuids,'dye assistants, insecticides and as plasticizers.for.vinyl resins. They-are also useful as plasticizers for celluloseacetate resins.

10 One half gram bfcommer'cial cellulose acetate which had beenthoroughly dried at 50- .C. was mixed in a 20 mm. x 120 mm. test-tubewith two grams of the compound to be tested. The test-tube was thenplaced in v a 200 C. heating 'bath for fifteen minutes. The appear:

ance of the mixture was observed while it was at 200 C. and after it hadcooled to room temperature.

The compounds'which weretested in the above manner were:tetr'a-(Z-chloroethyl) p,p-2-butylidenediplieiioxy-di-(chloropropyl)diphosphate; tetra-(2-chloroethyl) p phenylenedioxy di-(chloropropyl)diphosphate and tetra (2-chloroethyl)p,p'-sulfonyldiphenoxy-di-(chloropropyl) diphosphate, prepared,respectively in Examples 2, 4 and 6. Each of the compounds testedresulted in a mixture. which was .clear and colorless both at 200970.and'after, cooling; to room temperature. WhatI claim is: 1. A haloalkylaryl diphosphate having the formula:

in which X is a halogen" atom selected from the group consisting ofchlorine and bromine, Z is a member se'- lected from the classconsisting of p,p-lower alkylidene diphenyl, phenylene andp,p'-sulfonyldiphenyl radicals and Y is a phosphorus containing radicalselected from the class consisting of wherein R and R are membersselected from the class consisting of lower alkyl, lower chloroalkyl,lower bromoalkyl and monocyclic aryl radicals and. R? through R aremembers selected from the class consisting of hy; drogen and lower alkylgroups.

2. A haloalkyl aryl diphosphate having the formula? Kn i/POC2HsXCHz-OZO-CHz -C2HsX-O 1 R10 'OR;

in which X is a halogen atom selected from the group consisting ofchlorine and bromine, Z is a member s e v lected from the classconsisting of p,p'-lower alkylidenediphenyl, phenylene andp,p'.-sulfonylidphenyl radicals and R and R are members selected fromthe class consisting of lower alkyl, lower chloroalkyl, lower bromoalkyland monocyclic aryl radicals.

3. A haloalkyl aryl diphosphate having the formula:

in which X is a halogen atom selected from the. group consisting ofchlorine and bromine, Z is a member selected from the class consistingof p,p'-lower alkyliden'e- In order to demonstrate the compatibility ofthe com- 7 pounds of the present invention with cellulose acetate,several of the compounds were separately tested as follows:

diphenyl, phenylene and p,p'-sulfonyldipheny1 radicals and R through Rare members selected from the class consisting of hydrogen and loweralkyl groups.

0 0 II I II (CaH OhP O CaHsClCHaO-O-EOOCHzCfihCIO 1 (O C4H0)l 11 2A:tetra-- :(s2+chloroethyl) p,p'-2-butylidene-diphenoXy-di-(ehloropropyl)diphosphate represented by the structural formula:

CrHu :6. A tetra'(.2-chloroethyl)p,p'-ethylidenediphenoxydi-(eliloropropyl) diphosphate represented bythe struetura'l formula:

7. A tetra-'(Z-chloroethyl) p-phenylenedioxy-di-(chloropropyl)diphosphate represented by the structural formula:

8. A tetra-(.Z-chloroethyl) m-phenylenedioxy-di-(chloropropyl)ldiphosphate represented by the structural formula:

ll 4 .ll (ClCzHrOlnPOCaHzClCHiO -O OHnCaHaClO P (0 0911401):

9. A tetra-(Z-chloroethyl) p,p'-sulfonyldiphenoxy-di- 30 (chloropropyl)diphosphate represented by the structural formula:

propoxy) bis( 4-methyl-2-oxo- 1,3 ,2-dioxaphosphorinane) represented bythe structural formula:

14. The process which comprises bringing into admixture at a temperatureof from about 10 C. to about .10. A 2,2[p,p'-is0propylidenediphenoxy-di-(chloropropoxy)] bis(5,5 diethyl2-oxo-1,3,2-dioxaphosphorinane) represented by the structural formula:

2 11. A tetraphenyl p,p'-isopropylidenediphenoxy-di- (chloropropyl)diphosphate represented by the structural formula:

125 C. a diepoxy compound represented by the formula:

with a disubstitute'd 'monohalophosph'ate represented by the formula YX,wherein X is a halogen atom selected 45 from the group consisting ofchlorine and bromine, Z

is a member selected from the class consisting of p,p'- loweralkylidenediphen'yl, phenylene and p,p'-sulfonyldiphenyl radicals and Yis a phosphorus-containing radi- 5 12. A 2,2[p,p'-isopropylidenediphenoxy-di-(chloropropoxy)] bis(5ethyl-2-oxo-4-propyl-1,3,2-dioxaphosphorinane) represented by thestructural formula:

cal selected from the class consisting of No references cited UNITEDSTATES PATENT OFFICE Certificate of Correction Patent No. 2,880,225March 31, 1959 William M. Lanham It is hereby certified that errorappears in the printed specification of the above,

numbered patent requiring correction and that the said Letters Patentshould read as corrected below.

Column 1, line 62, for di-chloropropyl) read --di-(chloropropyl); column4, line 74;, in the footnotes for 7 6 grams read -7 .6 grams; column5,1ine 73, Example 5, for that portion of the formula reading column 6,line 59, Example 7, right-hand portion of the formula, for (OCJ-LCI)read (OO H Ol) column 8, line 9, after with insert water; column 10,line 49, for "p,p-sulfonylidpheny1 read -p,p'-sulfonyldiphenyl-.

Signed and sealed this 8th day of December 1959.

Attest: KARL H. AXLINE, Attestz'ng Oficer.

ROBERT C. WATSON, Uommz'ssioner of Patents.

1. A HALOAKYL ARYL DIPHOSPHATE HAVING THE FORMULA:
 14. THE PROCESS WHICHCOMPRISES BRINGING INTO ADMIXTURE AT A TEMPERATURE OF FROM ABOUT-10*C.TO ABOUT 125*C. A DIEPOXY COMPOUND REPRESENTED BY THE FORMULA